It is first useful to note that an impulse disturbance to the surface velocity, \({\partial h}{\partial t}\)  in Eq. (REF ), has the exact same effect as the impulse forcing function included in Eq. (REF ) (see REF  for details). For this reason, the initial velocity may be taken to be zero in Eq. (REF ) without loss of generality.
Additionally, an impulse disturbance to initial surface height, \(h\)  in Eq. (REF ), leads to a solution that violates the condition of \(h\rightarrow 0\)  as \(x,z\rightarrow \pm \infty \)  in Eq. (REF ); this nonphysical solution is included in REF . Although not stated explicitly in [1]}, this violation also occurs in 1D-KRK.
An interesting feature of this 2D nonphysical solution is that, because the Fourier transform of the solution exists through rapid oscillations of the sinusoidal integrand, the Fourier integral solution methodology admits the violation.
If one extends the class of allowable solutions to include spatially non-local responses, the solution nevertheless does damp with time. To avoid the nonphysical nature of this solution however, the initial height is chosen to be zero in Eq. (REF ). In summary, homogeneous constraints on the system in Eq. (REF ) are chosen without loss of generality in the stability conclusions that follow.
